3. A person somewhat versed in mathematics, having before him a series of numbers, will generally be able to devise a formula which approaches near to those numbers. If, for instance, the series is constantly progressive, he will be able to see whether it more nearly resembles an arithmetical or a geometrical progression. For example, MM. Dulong and Petit, in their investigation of the law of cooling of bodies, obtained the following series of measures. A thermometer, made hot, was placed in an enclosure of which the temperature was 0 degrees, and the rapidity of {197} cooling of the thermometer was noted for many temperatures. It was found that For the temperature 240 the rapidity of cooling was 10·69 220 " 8·81 200 " 7·40 180 " 6·10 160 " 4·89 140 " 3·88 and so on. Now this series of numbers manifestly increases with greater rapidity as we proceed from the lower to the higher parts of the scale. The numbers do not, however, form a geometrical series, as we may easily ascertain. But if we were to take the differences of the successive terms we should find them to be-- 1·88, 1·41, 1·30, 1·21, 1·01, &c. and these numbers are very nearly the terms of a geometric series. For if we divide each term by the succeeding one, we find these numbers, 1·33, 1·09, 1·07, 1·20, 1·27, in which there does not appear to be any constant tendency to diminish or increase. And we shall find that a geometrical series in which the ratio is 1·165, may be made to approach very near to this series, the deviations from it being only such as may be accounted for by conceiving them as errours of observation. In this manner a certain formula[26\3] is obtained, giving results {198} which very nearly coincide with the observed facts, as may be seen in the margin. [Note 26\3: The formula is _v_ = 2·037(_a^t_ - 1) where _v_ is the velocity of cooling, _t_ the temperature of the thermometer expressed in degrees, and _a_ is the quantity, 1·0077. The degree of coincidence is as follows:-- Excess of temperature of Observed Calculated the thermometer, or values values values of _t_. of _v_. of _v_. 240 10·69 10·68 220 8·81 8·89 200 7·40 7·34 180 6·10 6·03 160 4·89 4·87 140 3·88 3·89 120 3·02 3·05 100 2·30 2·33 80 1·74 1·72 ]